On homomorphisms between Cremona groups

Urech, Christian

doi:10.5802/aif.3151

On homomorphisms between Cremona groups
[Homomorphismes entre des groupes de Cremona]

Urech, Christian ¹

¹ IRMAR Université de Rennes 1 35042 Rennes (France)

Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 53-100.

Résumé
Abstract

On s’intéresse aux plongements algébriques du groupe de Cremona complexe à $n$ variables ${Cr}_{n}$ dans des groupes de transformations birationnelles $Bir (M)$ d’une varété algébrique $M$ . D’abord on regarde un plongement de ${Cr}_{2}$ dans ${Cr}_{5}$ qui était découvert par Gizatullin. Puis on donne une classification de tous les plongements algébriques de ${Cr}_{2}$ dans $Bir (M)$ pour des variétés $M$ de dimension $3$ et on généralise partiellement ce résultat aux plongements algébriques de ${Cr}_{n}$ dans $Bir (M)$ , où la dimension de $M$ est $n + 1$ (pour tout $n$ ). On obtient notamment une classification de toutes les action régulières de ${PGL}_{n + 1} (ℂ)$ sur des variétés projectives lisses de dimension $n + 1$ qui s’étendent vers des actions rationnelles de ${Cr}_{n}$ .

We look at algebraic embeddings of the complex Cremona group in $n$ variables ${Cr}_{n}$ to the group of birational transformations $Bir (M)$ of an algebraic variety $M$ . First we study geometrical properties of an example of an embedding of ${Cr}_{2}$ into ${Cr}_{5}$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of ${Cr}_{2}$ into $Bir (M)$ , where $M$ is a variety of dimension 3 and generalize this result partially to algebraic embeddings of ${Cr}_{n}$ into $Bir (M)$ , where the dimension of $M$ is $n + 1$ , for arbitrary $n$ . In particular, this yields a classification of all algebraic ${PGL}_{n + 1} (ℂ)$ -actions on smooth projective varieties of dimension $n + 1$ that can be extended to rational actions of ${Cr}_{n}$ .

Reçu le : 2016-04-25
Révisé le : 2016-12-05
Accepté le : 2017-01-24
Publié le : 2018-04-18

DOI : 10.5802/aif.3151

Classification : 14E07, 14L30, 32M05
Keywords: Cremona group, rational group actions, algebraic group actions
Mot clés : Groupe de Cremona, actions rationnelles des groupes

Affiliations des auteurs :

Urech, Christian ¹

¹ IRMAR Université de Rennes 1 35042 Rennes (France)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AIF_2018__68_1_53_0,
     author = {Urech, Christian},
     title = {On homomorphisms between {Cremona} groups},
     journal = {Annales de l'Institut Fourier},
     pages = {53--100},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3151},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3151/}
}

TY  - JOUR
AU  - Urech, Christian
TI  - On homomorphisms between Cremona groups
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 53
EP  - 100
VL  - 68
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3151/
DO  - 10.5802/aif.3151
LA  - en
ID  - AIF_2018__68_1_53_0
ER  -

%0 Journal Article
%A Urech, Christian
%T On homomorphisms between Cremona groups
%J Annales de l'Institut Fourier
%D 2018
%P 53-100
%V 68
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3151/
%R 10.5802/aif.3151
%G en
%F AIF_2018__68_1_53_0

Urech, Christian. On homomorphisms between Cremona groups. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 53-100. doi : 10.5802/aif.3151. https://aif.centre-mersenne.org/articles/10.5802/aif.3151/

Bibliographie
Cité par

[1] Akhiezer, Dmitri N. Lie group actions in complex analysis, Aspects of Mathematics, 27, Friedr. Vieweg & Sohn, 1995, viii+201 pages | DOI | Zbl

[2] Alberich-Carramiñana, Maria Geometry of the plane Cremona maps, Lecture Notes in Mathematics, 1769, Springer, 2002, xvi+257 pages | DOI | MR | Zbl

[3] Artamkin, I. V. Stable bundles with $c_{1} = 0$ on rational surfaces, Izv. Akad. Nauk SSSR Ser. Mat., Volume 54 (1990) no. 2, pp. 227-241 | Zbl

[4] Beauville, Arnaud; Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques; Swinnerton-Dyer, Peter Variétés stablement rationnelles non rationnelles, Ann. Math., Volume 121 (1985) no. 2, pp. 283-318 | DOI | MR | Zbl

[5] Białynicki-Birula, B. A. Remarks on the action of an algebraic torus on $k^{n}$ I, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., Volume 14 (1966), pp. 177-181 | Zbl

[6] Blanc, Jérémy Conjugacy classes of affine automorphisms of $𝕂^{n}$ and linear automorphisms of $ℙ^{n}$ in the Cremona groups, Manuscr. Math., Volume 119 (2006) no. 2, pp. 225-241 | DOI | MR | Zbl

[7] Blanc, Jérémy Sous-groupes algébriques du groupe de Cremona, Transform. Groups, Volume 14 (2009) no. 2, pp. 249-285 | DOI | MR | Zbl

[8] Blanc, Jérémy Symplectic birational transformations of the plane, Osaka J. Math., Volume 50 (2013) no. 2, pp. 573-590 http://projecteuclid.org/euclid.ojm/1371833501 | MR | Zbl

[9] Blanc, Jérémy; Déserti, Julie Degree growth of birational maps of the plane, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 14 (2015) no. 2, pp. 507-533 | MR | Zbl

[10] Blanc, Jérémy; Furter, Jean-Philippe Topologies and structures of the Cremona groups, Ann. Math., Volume 178 (2013) no. 3, pp. 1173-1198 | DOI | MR | Zbl

[11] Blanc, Jérémy; Hedén, Isac The group of Cremona transformations generated by linear maps and the standard involution, Ann. Inst. Fourier, Volume 65 (2015) no. 6, pp. 2641-2680 | DOI | Zbl

[12] Borel, Armand; Tits, Jacques Homomorphismes “abstraits” de groupes algébriques simples, Ann. Math., Volume 97 (1973), pp. 499-571 | DOI | MR | Zbl

[13] Brion, Michel Spherical Varieties: An Introduction, Topological methods in algebraic transformation groups (Progress in Mathematics), Volume 80, Springer, 1989, pp. 11-26 | Zbl

[14] Brunella, Marco Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. Éc. Norm. Supér., Volume 30 (1997) no. 5, pp. 569-594 | DOI | Zbl

[15] Cantat, Serge Endomorphismes des variétés homogènes, Enseign. Math., Volume 49 (2003) no. 3-4, pp. 237-262 | MR | Zbl

[16] Cantat, Serge Morphisms between Cremona groups, and characterization of rational varieties, Compos. Math., Volume 150 (2014) no. 7, pp. 1107-1124 | DOI | MR | Zbl

[17] Cantat, Serge; Zeghib, Abdelghani Holomorphic actions, Kummer examples, and Zimmer program, Ann. Sci. Éc. Norm. Supér., Volume 45 (2012) no. 3, pp. 447-489 | DOI | MR | Zbl

[18] Cerveau, Dominique; Déserti, Julie Birational maps preserving the contact structure on $ℙ_{ℂ}^{3}$ (2016) (http://arxiv.org/abs/1602.08866v1, to appear in J. Math. Soc. Japan)

[19] Corti, Alessio; Kaloghiros, Anne-Sophie The Sarkisov program for Mori fibred Calabi-Yau pairs, Algebr. Geom., Volume 3 (2016) no. 3, pp. 370-384 | DOI | Zbl

[20] Demazure, Michel Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 507-588 | DOI | MR | Zbl

[21] Déserti, Julie On the Cremona group: some algebraic and dynamical properties, Université Rennes 1 (France) (2006) https://tel.archives-ouvertes.fr/tel-00125492 (Theses)

[22] Déserti, Julie Sur les automorphismes du groupe de Cremona, Compos. Math., Volume 142 (2006) no. 6, pp. 1459-1478 | DOI | MR | Zbl

[23] Déserti, Julie Some properties of the group of birational maps generated by the automorphisms of $ℙ_{ℂ}^{n}$ and the standard involution (2015), pp. 893-905 (http://arxiv.org/abs/1403.0346v1) | Zbl

[24] Dieudonné, Jean A. La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 5, Springer, 1971, viii+129 pages | MR | Zbl

[25] Diller, Jeffrey; Lin, Jan-Li Rational surface maps with invariant meromorphic two-forms, Math. Ann., Volume 364 (2016) no. 1-2, pp. 313-352 | DOI | MR | Zbl

[26] Dolgachev, Igor V. Classical algebraic geometry. A modern view, Cambridge Univ. Press, 2012, xii+639 pages (A modern view) | DOI | MR | Zbl

[27] Enriques, Federigo Sui gruppi continui di trasformazioni cremoniane nel piano, Rend. Accad. Lincei, 1er sem, Volume 5 (1893) no. 1, pp. 468-473 | Zbl

[28] Fulton, William; Harris, Joe Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer, 1991, xvi+551 pages | DOI | Zbl

[29] Gizatullin, Marat On some tensor representations of the Cremona group of the projective plane, New trends in algebraic geometry (Warwick, 1996) (London Math. Soc. Lecture Note Ser.), Volume 264, Cambridge University Press, 1999, pp. 111-150 | DOI | MR | Zbl

[30] Gizatullin, Marat Klein’s conjecture for contact automorphisms of the three-dimensional affine space, Mich. Math. J., Volume 56 (2008) no. 1, pp. 89-98 | DOI | MR | Zbl

[31] de la Harpe, Pierre Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, 2000, vi+310 pages | MR | Zbl

[32] Hudson, Hilda P. Cremona transformation in Plane and Space, Cambridge University Press, 1927

[33] Humphreys, James E. Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer, 1975, xiv+247 pages | Zbl

[34] Lamy, Stéphane Une preuve géométrique du théorème de Jung, Enseign. Math., Volume 48 (2002) no. 3-4, pp. 291-315 | MR | Zbl

[35] Lamy, Stéphane On the genus of birational maps between threefolds, Automorphisms in birational and affine geometry (Springer Proceedings in Mathematics & Statistics), Volume 79, Springer, 2014, pp. 141-147 | DOI | MR | Zbl

[36] Pan, Ivan Une remarque sur la génération du groupe de Cremona, Bol. Soc. Bras. Mat., Nova Sér., Volume 30 (1999) no. 1, pp. 95-98 | DOI | MR | Zbl

[37] Popov, Vladimir L. Tori in the Cremona groups, Izv. Ross. Akad. Nauk Ser. Mat., Volume 77 (2013) no. 4, pp. 103-134 | MR | Zbl

[38] Procesi, Claudio Lie groups. An approach through invariants and representations, Universitext, Springer, 2007, xxiv+596 pages | MR | Zbl

[39] Serre, Jean-Pierre Local fields, Graduate Texts in Mathematics, 67, Springer, 1979, viii+241 pages (Translated from the French by Marvin Jay Greenberg) | MR | Zbl

[40] Serre, Jean-Pierre Le groupe de Cremona et ses sous-groupes finis, Séminaire Bourbaki 2008/2009 (Astérisque), Volume 332, Société Mathématique de France, 2010, pp. 75-100 | Zbl

[41] Stampfli, Immanuel A note on automorphisms of the affine Cremona group, Math. Res. Lett., Volume 20 (2013) no. 6, pp. 1177-1181 | DOI | MR | Zbl

[42] Sumihiro, Hideyasu Equivariant completion, J. Math. Kyoto Univ., Volume 14 (1974), pp. 1-28 | DOI | MR | Zbl

[43] Umemura, Hiroshi Maximal algebraic subgroups of the Cremona group of three variables. Imprimitive algebraic subgroups of exceptional type, Nagoya Math. J., Volume 87 (1982), pp. 59-78 http://projecteuclid.org/euclid.nmj/1118786899 | DOI | MR | Zbl

[44] Umemura, Hiroshi On the maximal connected algebraic subgroups of the Cremona group. I, Nagoya Math. J., Volume 88 (1982), pp. 213-246 http://projecteuclid.org/euclid.nmj/1118787013 | DOI | MR | Zbl

[45] Umemura, Hiroshi On the maximal connected algebraic subgroups of the Cremona group. II, Algebraic groups and related topics (Kyoto/Nagoya, 1983) (Adv. Stud. Pure Math.), Volume 6, North-Holland, Amsterdam, 1985, pp. 349-436 | MR | Zbl

[46] Urech, Christian Subgroups of Cremona groups, University of Basel (Switzerland) / University of Rennes 1 (France) (2017) (Ph. D. Thesis)

[47] Weil, André On algebraic groups of transformations, Am. J. Math., Volume 77 (1955), pp. 355-391 | DOI | Zbl

[48] Williams, A. R. Birational transformations in 4-space and 5-space, Bull. Am. Math. Soc., Volume 44 (1938) no. 4, pp. 272-278 | DOI | MR | Zbl

[49] Zaitsev, Dmitri Regularization of birational group operations in the sense of Weil, J. Lie Theory, Volume 5 (1995) no. 2, pp. 207-224 | Zbl

[50] Zhang, De-Qi The $g$ -periodic subvarieties for an automorphism $g$ of positive entropy on a compact Kähler manifold, Adv. Math., Volume 223 (2010) no. 2, pp. 405-415 | DOI | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT